Molecules: What Kind of a Bag of Atoms?

As discussed by Liang and Dill, Enright and Leitner, and others, proteins are not 3D objects. We study an expanded macromolecular data set ranging from proteins to RNA, lipids, and viruses, and remove surface effects and size bias. Molecules and molecular assemblies with more than 1000 backbone atoms have a volume fractal dimension of D-nu = 2.70 +/- 0.05 by the embedded sphere method and D-nu = 2.71 +/- 0.04 by the ensemble method using radius of gyration as the size measure. The much larger D-nu = 2.89 +/- 0.05 obtained with the average surface radius as the length measure shows that surface corrugation is as extensive as cavity formation. Using a simple "Swiss cheese" model for molecules, we show that the distribution of voids in the interior of molecules cannot be a Boltzmann distribution of void energy as a function of void size. Instead, frustration from imperfect packing builds up with molecular size, allowing larger voids to form in larger molecules. We find that large molecules lie halfway between the extremes of packing for homogeneous objects (D = 3) and Apollonian packing, which accounts for packing of a hierarchy of random-sized objects (D approximate to 2.47).